##
**Theory of nonlinear lattices.
2nd enlarged edition.**
*(English)*
Zbl 0694.70001

Springer Series in Solid-State Sciences, 20. Berlin etc.: Springer-Verlag. x, 225 p. DM 60.00 (1989).

The book should be called more appropriately “The Toda Lattice”. The bulk of the book is devoted to the solutions of the “classical” Toda lattice (the periodic and the infinite versions). In my opinion, the scope of the book is too narrow.

The presentation is big on explicit formulas and rather small on the conceptual side, which makes the book hard to read for a mathematician (or anybody else for that matter) not very familiar with the subject. On the other hand, the book is not addressed to an expert either. It is the second English edition [see the review of the first 1981 ed. in Zbl 0465.70014], of the Japanese original which appeared in 1978 (Zbl 0502.70013). The numerous developments in the integrable Hamiltonian systems that took place in the past decade are left out. As the author says in the preface, “a new chapter has been added to describe several important recent achievements”. Unfortunately, the addition (chapter 6: Recent advances in the theory of nonlinear lattices) falls short of this goal. It barely touches on a few subjects related to the Toda lattice and does not do justice to any of them. Suffices to say that chapter 6 is 18 pages long and consists of 9 subsections each devoted to a separate topic. The topics vary from the KdV equation to the “Integrability” to the “Bethe Ansatz” to the “Numerical Results”. The book would have been better without it.

The presentation is big on explicit formulas and rather small on the conceptual side, which makes the book hard to read for a mathematician (or anybody else for that matter) not very familiar with the subject. On the other hand, the book is not addressed to an expert either. It is the second English edition [see the review of the first 1981 ed. in Zbl 0465.70014], of the Japanese original which appeared in 1978 (Zbl 0502.70013). The numerous developments in the integrable Hamiltonian systems that took place in the past decade are left out. As the author says in the preface, “a new chapter has been added to describe several important recent achievements”. Unfortunately, the addition (chapter 6: Recent advances in the theory of nonlinear lattices) falls short of this goal. It barely touches on a few subjects related to the Toda lattice and does not do justice to any of them. Suffices to say that chapter 6 is 18 pages long and consists of 9 subsections each devoted to a separate topic. The topics vary from the KdV equation to the “Integrability” to the “Bethe Ansatz” to the “Numerical Results”. The book would have been better without it.

Reviewer: E.Gutkin

### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70Jxx | Linear vibration theory |

82B05 | Classical equilibrium statistical mechanics (general) |